correctionkey=nl-b;ca-b name class date 15.3 tangents and circumscribed...

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Explore Investigating the Tangent-Radius TheoremA tangent is a line in the same plane as a circle that intersects the circle in exactly one point. The point where a tangent and a circle intersect is the point of tangency.

In the figure, the line is tangent to circle C, and point P is the point of tangency. You can use a compass and straightedge to construct a circle and a line tangent to it.

A Use a compass to draw a circle. Label the center C.

B Mark a point P on the circle. Using a straightedge, draw a tangent to the circle through point P. Mark a point Q at a different position on the tangent line.

C Use a straightedge to draw the radius _ CP .

D Use a protractor to measure ∠CPQ. Record the result in the table. Repeat the process two more times. Make sure to vary the size of the circle and the location of the point of tangency.

Vehicle 1 Circle 1 Circle 2 Circle 3

Measure of ∠CPQ

Reflect

1. Make a Conjecture Examine the values in the table. Make a conjecture about the relationship between a tangent line and the radius to the point of tangency.

2. Discussion Describe any possible inaccuracies related to the tools you used in this Explore.

Module 15 805 Lesson 3

15.3 Tangents and Circumscribed Angles

Essential Question: What are the key theorems about tangents to a circle?

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Explain 1 Proving the Tangent-Radius TheoremThe Explore illustrates the Tangent-Radius Theorem.

Tangent-Radius Theorem

If a line is tangent to a circle, then it is perpendicular to a radius drawn to the point of tangency.

Example 1 Complete the proof of the Tangent–Radius Theorem.

Given: Line m is tangent to circle C at point P.

Prove: CP ⊥ m

A Use an indirect proof. Assume that CP is not perpendicular to line m. There must be a point Q on line m such that CQ ⊥ m.

If CQ ⊥ m, then △CQP is a triangle, and CP > CQ because

is the of the right triangle.

B Since line m is a tangent line, it can intersect circle C at only point , and all other points

of line m are in the of the circle.

C This means point Q is in the of the circle. You can conclude that CP < CQ because

is a of circle C.

D This contradicts the initial assumption that a point Q exists such that _ CQ ⊥ m, because that meant that CP > CQ. Therefore, the assumption

is and must be perpendicular to line m.

Reflect

3. Both lines in the figure are tangent to the circle, and _ AB is a diameter. What can

you conclude about the tangent lines?


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The converse of the Tangent-Radius Theorem is also true. You will be asked to prove this theorem as an exercise.

Converse of the Tangent-Radius Theorem

If a line is perpendicular to a radius of a circle at a point on the circle, then it is tangent to the circle at that point on the circle.

Explain 2 Constructing Tangents to a CircleFrom a point outside a circle, two tangent lines can be drawn to the circle.

Example 2 Use the steps to construct two tangent lines from a point outside a circle.

A Use a compass to draw a circle. Label the center C.

B Mark a point X outside the circle and use a straightedge to draw

_ CX .

C Use a compass and straightedge to construct the midpoint of

_ CX and label the midpoint M.

D Use a compass to construct a circle with center M and radius CM.

Label the points of intersection of circle C and circle M as A and B. Use a straightedge to draw

‹ − › XA and XB. Both lines are tangent to circle C.

Reflect

4. How can you justify that ‹ −

› XA (or XB) is a tangent line? (Hint: Draw _ CA on the diagram.)

5. Draw _ CA and

_ CB on the diagram. Consider quadrilateral CAXB. State any conclusions

you can reach about the measures of the angles of CAXB.

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Explain 3 Proving the Circumscribed Angle TheoremA circumscribed angle is an angle formed by two rays from a common endpoint that are tangent to a circle.

Circumscribed Angle Theorem

A circumscribed angle of a circle and its associated central angle are supplementary.

Example 3 Prove the Circumscribed Angle Theorem.

Given: ∠AXB is a circumscribed angle of circle C.

Prove: ∠AXB and ∠ACB are supplementary.

Since ∠AXB is a circumscribed angle of circle C, _ XA and

_ XB are

to the circle. Therefore, ∠XAC and ∠XBC are

by the .

In quadrilateral XACB, the sum of the measures of its four angles is .

Since m∠XAC + m∠XBC = , this means m∠AXB + m∠ACB = 360° - 180° = .

So, ∠AXB and ∠ACB are supplementary by the .

Reflect

6. Is it possible for quadrilateral AXBC to be a parallelogram? If so, what type of parallelogram must it be? If not, why not?

Elaborate

7. ‾→ KM and

‾→ KN are tangent to circle C. Explain how to show that _

KM ≅ _ KN , using congruent triangles.

8. Essential Question Check-In What are the key theorems regarding tangent lines to a circle?



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• Online Homework• Hints and Help• Extra Practice

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SAEvaluate: Homework and Practice

Use the figure for Exercises 1–2. You use geometry software to construct a tangent to circle O at point X on the circle, as shown in the diagram.

1. What do you expect to be the measure of ∠OXY? Explain.

2. Suppose you drag point X so that is in a different position on the circle. Does the measure of ∠OXY change? Explain.

3. Make a Conjecture You use geometry software to construct circle A, diameters

_ AB and

_ AD , and lines m and n which are

tangent to circle A at points D and B, respectively. Make a conjecture about the relationship of the two tangents. Explain your conjecture.

4. In the figure, _ RQ is tangent to circle P at point Q. What is m∠PRQ?

Explain your reasoning.

5. Represent Real-World Problems The International Space Station orbits Earth at an altitude of about 240 miles. In the diagram, the Space Station is at point E. The radius of Earth is approximately 3960 miles. To the nearest ten miles, what is EH, the distance from the space station to the horizon?



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Multi-Step Find the length of each radius. Identify the point of tangency, and write the equation of the tangent line at that point.

6. 7.

8. In the figure, QS = 5, RT = 12, and ‹ −

› RT is tangent to radius _ QR with the point of

tangency at R. Find QT.

The segments in each figure are tangent to the circle at the points shown. Find each length.

9. 10.



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11. Justify Reasoning Suppose you construct a figure with _ PR tangent to

circle Q at R and _ PS tangent to circle Q at S. Make a conjecture about ∠P

and ∠Q . Justify your reasoning.

12. _ PR is tangent to circle Q at R and

_ PS is tangent to circle Q at S. Find m∠Q.

13. _ PR is tangent to circle Q at R and

_ PS is tangent to circle Q at S. Find m∠P.

PA is tangent to circle O at A and PB is tangent to circle O at B, and m∠P = 56°. Use the figure to find each measure.

14. m∠AOB

15. m∠OGF



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16. Which statements correctly relate ∠BDC and ∠BAC? Select all that apply.

A. ∠BDC and ∠BAC are complementary.

B. ∠BDC and ∠BAC are supplementary.

C. ∠BDC and ∠BAC are congruent.

D. ∠BDC and ∠BAC are right angles.

E. The sum of the measures of ∠BDC and ∠BAC is 180°.

F. It is impossible to determine a relationship between ∠BDC and ∠BAC.

17. Critical Thinking Given a circle with diameter _ BC , is it possible to construct

tangents to B and C from an external point X? If so, make a construction. If not, explain why it is not possible.

‾→ KJ is tangent to circle C at J,

‾→ KL is tangent to circle C at L,

and m ⁀ ML = 138°.

18. Find m∠M.

19. Find m ⁀ MJ .



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H.O.T. Focus on Higher Order Thinking

20. Justify Reasoning Prove the converse of the Tangent-Radius Theorem. Given: Line m is in the plane of circle C, P is a point of circle C, and

_ CP ⊥ m

Prove: m is tangent to circle C at P.

21. Draw Conclusions A grapic designer created a preliminary sketch for a company logo. In the figure,

‹ − › BC and

‹ − › CD are tangent to circle A and

BC > BA. What type of quadrilateral is figure ABCD that she created? Explain.

22. Explain the Error In the given figure, ‹ −

› QP and ‹ −

› QR are tangents. A student was asked to find m∠PSR. Critique the student’s work and correct any errors.

Since ∠PQR is a circumscribed angle, ∠PQR and ∠PCR are supplementary. So m∠PCR = 110°. Since ∠PSR ≅ ∠PQR, m∠PCR = 110°.

23. Given circle O and points B and C, construct a triangle that is circumscribed around the circle.



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Lesson Performance Task

A communications satellite is in a synchronous orbit 22,000 miles above Earth’s surface. Points B and D in the figure are points of tangency of the satellite signal with the Earth. They represent the greatest distance from the satellite at which the signal can be received directly. Point C is the center of the Earth.

1. Find distance AB. Round to the nearest mile. Explain your reasoning.

2. m∠BAC = 9°. If the circumference of the circle represents the Earth’s equator, what percent of the Earth’s equator is within range of the satellite’s signal? Explain your reasoning.

3. How much longer does it take a satellite signal to reach point B than it takes to reach point E? Use 186,000 mi/sec as the speed of a satellite signal. Round your answer to the nearest hundredth.

4. The satellite is in orbit above the Earth’s equator. Along with the point directly below it on the Earth’s surface, the satellite makes one complete revolution every 24 hours. How fast must it travel to complete a revolution in that time? You can use the formula C = 2πr to find the circumference of the orbit. Round your answer to the nearest whole number.

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